A consumer h$,$ where h $=1$; $2,$ wishes to maximise her utility

max Uh $=1$ ln$($xh$)+0$:$51$ ln$($G$)(1)$

xh ;gh subject to the budget constraint

Mh$=$ xh$+$gh$,(2)$ G $=$ g$1+$ g$2.(3)$

where

In this speciO$$cation$,$ ln denotes the natural logarithm, xh denotes a private good

and gh denotes the contribution of each consumer to the public good G$.$

$(3\mathrm{.}1).$ Explain the meaning of all parameters that appear in the optimisation problems of the two consumers. Calculate, for consumer $1,$ the marginal rate of substitution and the marginal rate of transformation between the two goods, expressed in terms of g$1,$ and explain what they mean in this context. $[10\%]$

$(3\mathrm{.}2).$ Show that the utility maximising provision of the public good for consumer $1$ is decreasing in the quantity of the good provided by the other consumer, and explain the logic for this behaviour. Show and explain the logic of all steps in your work. $[15\%]$

$(3\mathrm{.}3).$ Show that the market equilibrium, where each consumer chooses their contribution to the public good, implies equalisation of the marginal rate of

$1$

Continued Overleaf

substitution with the marginal rate of transformation for each consumer. Show and explain the logic of all steps in your work. $[10\%]$

$(3\mathrm{.}4).$ Show that the quantity of the public good, G$,$ must increase from the level provided by the consumers in the market equilibrium in order to achieve the e$$ cient quantity associated with the Samuelson rule. Show and explain the logic of all steps in your work. $[15\%]$